Vol 2, No 11 (2011)

RANDOM FIXED POINT THEOREMS OF RANDOM

MULTIVALUED OPERATORS ON POLISH SPACE

Rajesh shrivastava*, Animesh Gupta* and R. N. Yadav**

*Department of Mathematics, Govt. Benazir Science & Commerce College, Bhopal, M.P., India

**Ex-Director Gr. Scientist & Head, Resource Development Center, Regional Research Laboratory, Bhopal-462016, India

E-mail:[email protected]

(Received on: 16-10-11; Accepted on: 01-11-11)

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ABSTRACT

I

Keywords: Polish Space, Random Operator, Random Multivalued Operator, Random Fixed Point, Measurable

mapping.

AMS Subject Classification: 47H10, 54H25.

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1. INTRODUCTION:

Probabilistic functional analysis has emerged as one of the important mathematical disciplines in view of its role in analyzing Probabilistic models in the applied sciences. The study of fixed point of random operator forms a central topic in this area. Random fixed point theorem for contraction mappings in Polish spaces and random Fixed point theorems are of fundamental importance in probabilistic functional analysis. There study was initiated by the Prague school of Probabilistics, in 1950, with their work of Spacek [15] and Hans [5,6]. For example survey is refer to Bharucha-Reid [4]. Itoh [8] proved several random fixed point theorems and gave their applications to Random differential equations in Banach spaces. Random coincidence point theorems and random fixed point theorems are stochastic generalization of classical coincidence point theorems and classical fixed point theorems. Sehgal and Singh [14], Papageorgiou [12], Rhoades, Sessa, Khan [13] and Lin [11] have proved differential Stochastic version of well known Schauder’s fixed point theorem. Recently, Beg and Shahzad [3] studied the structure of common fixed point and random coincidence Points of a pair of compatible random operators.

2. PRELIMINARIES:

Definition: 2.1 A metric space is said to be a Polish Space, if it satisfying following conditions:

(1) X, is complete, (2) X is separable,

Before we describe our next hierarchy of set of reals of ever increasing complexity,

we would like to consider a class of metric spaces under which we can unify and there products. This will be helpful in formulating this hierarchy

Recall that a metric space is complete if whenever is a sequence of member of X, such that for every there is an such that implies _! " , there is a single # such that $%& _' ( . It is easy to see that are polish space, So in fact is under the discreate topology, whose metric is given by letting ) ( * when + ) and ) ( when ( )

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$ % ! & % & ' ( Let be a Polish space that is a separable complete metric space and , be Measurable space. Let - be a family of all subsets of and ./ denote the family of all nonempty bounded closed subsets of A mapping 0 1 2 is called measurable if for any open subset . of , 034 . ( 5 6 . +

, A mapping 7 is said to be measurable selector of a measurable mapping 0 , if 7 is measurable and for any 7 0 . A mapping 8 1 is called random operator, if for any

9 is measurable. A Mapping 0 8 1 ./ is a random multivalued operator, if for every

0 9 is measurable . A measurable mapping 7 1 is called random fixed point of a random multivalued operator 0 8 1 ./ 8 1 if for every 7 0: 7 ; 7 (

7 Let 0 8 1 ./ be a random operator And 57 < a sequence of measurable mappings , 7 1 The sequence 57 < is said to be asymptotically T-regular if 7 0: 7 ; 1

3. MAIN RESULT:

Theorem 3.1: Let be a Polish space. Let 0 = > 8 1 ./ be two continuous random multivalued operators, If there exists measurable mappings ? > 1 *, such that, it satisfy followings condition,

@:= 0 ) ; A B :) 0 ) ; : = ; ) C D

E _{F * *}

For each ) and GH # I " * Then there exists a common random fixed point of

= 0. @ @ J I @ #K ./ # K ?) I #K

Proof: Let 7L> 1 be an arbitrary measurable mapping and choose a measurable mapping 7 1 such that 74 =: 7L ; for each then for each

@ M=: 7L ; 0: 74 ;N A O M74 0: 74 ;N M7L =: 7L ;N :7L 74 ; P

D E

Further, there exists a measurable mapping 7Q> 1 such that for all 7Q 0 74 and,

:74 7Q ; ( @:= 7L 0 74 ;

:74 7Q ; A B :74 7Q ; :7L 74 ; :7L 74 ;CDE

:74 7Q ; A : ;ER _{7L 74}

Let S ( : ; E

R _{" * then,}

:74 7Q ; A S 7L 74

By Beg and Shahzad [2, lemma2.3], we obtain a measurable mapping 7T > 1 such that for all 7T =: 7Q ; and,

:7Q 7T ; ( @:= 74 0 7Q ;

:7Q 7T ; A B :7Q 7T ; :74 7Q ; :74 7Q ;CDE

:7Q 7T ; A S 74 7Q

:7Q 7T ; A SQ _{7L 74}

Similarly, proceeding in the same way: by induction, we produce a sequence of measurable mapping

7 > 1 , such that for and any , 7Q H4 =: 7Q ; , 7Q HQ 0: 7Q H4 ; and hence,

:7 7 H4 ; A S 7L 74

$ % ! & % & '

:7 7! ; A :7 7 H4 ; U :7 H4 7 HQ ; U

V V V U :7!34 7! ;

:7 7! ; A WS U S H4_{U V V U S}!34_{X :7L 74 ;}

:7 7! ; A W* U S U V V U S!3 34_{X S :7L 74 ;}

:7 7! ; A Y_43ZZ[\ :7L 74 ;

:7 7! ; 1 1

It follows that, 57 < is a Cauchy sequence and there exists a measurable mapping 7 > 1 such that 7 1 7 , for each It further implies that, 7Q H4 1 7 and 7Q HQ 1 7 . Thus we have for any

M7 =: 7 ;N A :7 7Q HQ ; U M7Q HQ =: 7 ;N

M7 =: 7 ;N A :7 7Q HQ ; U @:= 7 0 7Q H4 ;

By using F * * 1 we have

M7 =: 7 ;N A

Hence 7 =: 7 ; for similarly, for any

M7 0: 7 ;N A :7 7Q H4 ; U M7Q H4 0: 7 ;N

M7 0: 7 ;N A :7 7Q H4 ; U @:= 7Q 0 7 ;

By using F * * 1 we have

M7 0: 7 ;N A

Hence 7 0: 7 ; for similarly, for any

Theorem 3.2: Let be a Polish space. Let 0 = > 8 1 ./ be two continuous random multivalued operators, If there exists measurable mappings ? > 1 *, such that, it satisfy followings condition,

@:= 0 ) ; A &]^B ) :) 0 ) ; : = ;C U ? B : 0 ) ; U :) = ;C

F *

For each ) and ? GH # I U ? " * Then there exists a common random fixed

point of = 0.@ @ J I @ #K ./ # K ?) I #K

Proof: Let 7L> 1 be an arbitrary measurable mapping and choose a measurable mapping 7 1 such that 74 =: 7L ; for each then for each

@ M=: 7L ; 0: 74 ;N A &]^ _ :7L 74 ; M74 0: 74 ;N

M7L =: 7L ;N `

U ? O M7L 0: 74 ;N U M74 =: 7L ;NP

Further, there exists a measurable mapping 7Q> 1 such that for all 7Q 0 74 and,

$ % ! & % & ' )

:74 7Q ; A &]^ _ :7L 74 ; M74 0: 74 ;N

M7L =: 7L ;N `

U ? O M7L 0: 74 ;N U M74 =: 7L ;NP

:74 7Q ; A &]^ a :7L 74_:7L ; :74_{74 ;}7Q ;b

U ? B :7L 7Q ; U :74 74 ;C

:74 7Q ; A W U ? X 7L 74

Let S ( W U ? X " * then,

:74 7Q ; A S 7L 74

By Beg and Shahzad [lemma2.3], we obtain a measurable mapping 7T > 1 such that for all 7T =: 7Q ; and,

:7Q 7T ; ( @:= 74 0 7Q ;

:74 7Q ; A &]^ a :74 7Q_:74 ; :7Q_{7Q ;} 7T ;b

U ? B :74 7T ; U :7Q 7Q ;C

:7Q 7T ; A W U ? X 74 7Q

Let S ( W U ? X " * then,

:7Q 7T ; A SQ _{7L 74}

Similarly, proceeding in the same way: by induction, we produce a sequence of measurable mapping

7 > 1 , such that for and any , 7Q H4 =: 7Q ; , 7Q HQ 0: 7Q H4 ;

And hence,

:7 7 H4 ; A S 7L 74

Furthermore, for

:7 7! ; A :7 7 H4 ; U :7 H4 7 HQ ; U

V V V U :7!34 7! ;

:7 7! ; A WS U S H4_{U V V U S}!34_{X :7L 74 ;}

:7 7! ; A W* U S U V V U S!3 34_{X S :7L 74 ;}

:7 7! ; A Y_43ZZ[\ :7L 74 ;

:7 7! ; 1 1

It follows that, 57 < is a Cauchy sequence and there exists a measurable mapping 7 > 1 such that 7 1 7 , for each It further implies that, 7Q H4 1 7 and 7Q HQ 1 7 . Thus we have for any

$ % ! & % & ' *

M7 =: 7 ;N A :7 7Q HQ ; U @:= 7 0 7Q H4 ;

By using F * and As 1 we have

M7 =: 7 ;N A

Hence 7 =: 7 ; for similarly, for any

M7 0: 7 ;N A :7 7Q H4 ; U M7Q H4 0: 7 ;N

M7 0: 7 ;N A :7 7Q H4 ; U @:= 7Q 0 7 ;

As 1 we have

M7 0: 7 ;N A

Hence 7 0: 7 ; for similarly, for any

Theorem 3.3: Let be a Polish space. Let 0 = > 8 1 ./ be two continuous random multivalued operators, If there exists measurable mappings ? K > 1 *, such that, it satisfy followings condition,

@:= 0 ) ; A c :) = ; U : 0 ) ;d

U ? c :) 0 ) ; U : = ;d U K ) F * F

For each ) and e ? K GH # I U ? U K " * " f_{4 3 f}H g_3gHh " * Then there exists a common random fixed point of = 0 .

@ @ J I @ #K ./ # K ?) I #K

Proof: Let 7L> 1 be an arbitrary measurable mapping and choose a measurable mapping 7 1 such that 74 =: 7L ; for each then for each

@ M=: 7L ; 0: 74 ;N A O M74 =: 7L ;N U M7L 0: 74 ;NP

U ? O M74 0: 74 ;N U M7L =: 7L ;NP U K 7L 74

Further, there exists a measurable mapping 7Q> 1 such that for all 7Q 0 74 and,

:74 7Q ; ( @:= 7L 0 74 ;

:74 7Q ; A O M74 =: 7L ;N U M7L 0: 74 ;NP

U ? O M74 0: 74 ;N U M7L =: 7L ;NP U K 7L 74

:74 7Q ; A B :74 74 ; U :7L 7Q ;C

U ? B :74 7Q ; U :7L 74 ;CU K 7L 74

:74 7Q ; A Yf_{4 3 f}H g_3gHh \ 7L 74

Let S ( Yf_{4 3 f}H g_3gHh \ " * then,

:74 7Q ; A S 7L 74

$ % ! & % & ' +

:7Q 7T ; ( @:= 74 0 7Q ;

:7Q 7T ; A B :7Q 7Q ; U :74 7T ;C

U ? B :7Q 7T ; U :74 7Q ;CU K 74 7Q

:7Q 7T ; A Yf_{4 3 f}H g_3gHh \ 74 7Q

Let S ( Yf_{4 3 f}H g_3gHh \ " * then,

:7Q 7T ; A SQ _{7L 74}

Similarly, proceeding in the same way: by induction, we produce a sequence of measurable mapping

7 > 1 , such that for and any , 7Q H4 =: 7Q ; , 7Q HQ 0: 7Q H4 ;

And hence,

:7 7 H4 ; A S 7L 74

Furthermore, for

:7 7! ; A :7 7 H4 ; U :7 H4 7 HQ ; U

V V V U :7!34 7! ;

:7 7! ; A WS U S H4_{U V V U S}!34_{X :7L 74 ;}

:7 7! ; A W* U S U V V U S!3 34_{X S :7L 74 ;}

:7 7! ; A Y_43ZZ[\ :7L 74 ;

:7 7! ; 1 1

It follows that, 57 < is a Cauchy sequence and there exists a measurable mapping 7 > 1 such that 7 1 7 , for each It further implies that, 7Q H4 1 7 and 7Q HQ 1 7 . Thus we have for any

M7 =: 7 ;N A :7 7Q HQ ; U M7Q HQ =: 7 ;N

M7 =: 7 ;N A :7 7Q HQ ; U @:= 7 0 7Q H4 ;

By using F * F and As 1 we have

M7 =: 7 ;N A

Hence 7 =: 7 ; for similarly, for any

M7 0: 7 ;N A :7 7Q H4 ; U M7Q H4 0: 7 ;N

M7 0: 7 ;N A :7 7Q H4 ; U @:= 7Q 0 7 ;

As 1 we have

M7 0: 7 ;N A

$ % ! & % & ' #

ACKNOWLEDGEMENT:

We Heartily gratefull to Dr. S.S, Rajput Head of Department, (Mathematics), Govt. P. G Collage, Gadarwara for the motivation and Guidance during the preparation of this article.

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